1. Introduction · to a Kurosh-Amitsur prime radical for nearrings (see [13]). Veljko [37,38] gave...

MATEMATIČKI VESNIK

MATEMATIQKI VESNIK

70, 1 (2018), 64–78

March 2018

research paper

originalni nauqni rad

ΘΓ N-GROUP

Hamsa Nayak, Syam Prasad Kuncham and Babushri Srinivas Kedukodi

Abstract. In this paper, we introduce the notion of ΘΓ N -group as a generalization ofalgebraic structures of N -group and gamma nearring. We present motivating examples ofΘΓ N -groups and prove classical isomorphism theorems.

1. Introduction

A nearring (N,+, ·) is an algebraic system with binary operations addition and mul-tiplication satisfying the axioms of a ring, except commutativity of addition and oneof the distributive laws. A natural example of nearring is the set of all mappingsfrom a group (G,+) to itself under addition and composition of mappings. Groe-newald [18], Veldsman [36] introduced different types of prime ideals of nearringssuch as completely prime, 3-prime and equiprime ideals. Equiprime ideals gave riseto a Kurosh-Amitsur prime radical for nearrings (see [13]). Veljko [37, 38] gave defi-nitions of nilpotency, nilty, nil-radical, nilpotent-radical and nearring homomorphismof a general (non associative and non distributive) nearring and studied its affineendomorphism. N -groups are modules over nearrings (see [33]). Juglal, Groenewaldand Lee [22] introduced characterizations of prime modules of zero symmetric near-ring. Groenewald, Juglal and Meyer [19] discussed relations between primeness of zerosymmetric nearring and its group nearring. Nobusawa [32] introduced Γ-ring, a gen-eralization of ring. Barnes [3] studied notions of Γ-homomorphism, prime and (right)primary ideals, m-systems, radical of an ideal in Γ-rings. Sapanci and Nakajimaz [35]gave the condition for commutative property in gamma rings. Bell and Argac [4]studied derivations, product of derivations in nearrings and obtained commutativityresults under suitable conditions.

Bhavanari [7] introduced gamma nearrings, a generalization of both nearrings andgamma-rings. This concept was further studied in [5, 6, 11, 27] and several resultswere proved. Booth and Groenewald [12, 14] introduced equiprime gamma nearrings

2010 Mathematics Subject Classification: 16Y30

Keywords and phrases: N -group, nearring, gamma nearring.

64

H. Nayak, S. P. Kuncham, B. S. Kedukodi 65

and radicals of gamma nearrings. Jun, Sapanci and Ozturk [23] studied fuzzy idealsof gamma nearrings. Bhavanari and Kuncham [10] introduced the notion of a fuzzycoset in gamma nearring and obtained related important fundamental isomorphismtheorems. Booth, Groenewald and Olivier [15] defined general regularity for gammarings and explored ways of generating such regularities. Kedukodi, Kuncham andBhavanari [24, 25] studied equiprime, 3-prime and c-prime fuzzy ideals of nearrings.As an application of equiprime ideals, in [26] the notion of reference point in roughsets was introduced. In [8], the same authors also studied graph theoretic aspectsnearrings. Jagadeesha, Kedukodi, Kuncham [21] defined interval valued L-fuzzy idealsof nearrings based on t-norms and t-conorms and in [28], they studied homomorphicimages of interval valued L-fuzzy ideals and proved isomorphism theorems.

In this paper, we introduce the notion of ΘΓ N -group which is a generalizationof N -group and gamma nearring. A ΘΓ N -group is an algebraic structure wherethe operations belonging to the set Θ satisfy the right distributive property and thequasi associative property. We place on record the starting step where the idea of ΘΓN -group arose. In the real number system, we know that the operations of subtrac-tion and division are not associative. This is unlike their respective counterparts ofaddition and multiplication. However, we note that the operations subtraction anddivision are near associative operations. Consider the abelian group (R,+) and takea, b, c ∈ R. Corresponding to usual subtraction, we can define an operation “subc” bya subc b = (a− 2c)− b. Then we have (a− b)− c = a subc(b− c). We name this nearassociativity as the quasi associative property. In Example 3.6 of this paper, we showthat a similar quasi associative property is satisfied by usual division operation.

2. Preliminaries

We refer to [30,31] for basic definitions, and for recent developments in nearrings, werefer to [29]. Computations in nearrings can be done using SONATA [1].

Definition 2.1. [33] Let (G,+) be a group with additive identity 0. G is said tobe an N -group if there exist a nearring (N,+, ·), and a mapping N × θ × G → G(the image (n, g) ∈ N × θ ×G is denoted by nθg where θ is an operation), satisfying(n + m)θg = nθg + mθg and (nm)θg = nθ(mθg), for all g ∈ G and n,m ∈ N . Wedenote this N -group by NG.

Definition 2.2. [9] Let (M,+) be a group (not necessarily abelian) and Γ be a non-empty set. Then M is said to be a Γ-nearring if there exists a mappingM ×Γ×M →M (denote the image of (m1, α1,m2) by m1α1m2 for m1,m2 ∈M andα1 ∈ Γ) satisfying the following conditions:(m1 +m2)α1m3 = m1α1m3 +m2α1m3 and (m1α1m2)α2m3 = m1α1(m2α2m3),

for all m1,m2,m3 ∈M and for all α1, α2 ∈ Γ.

Definition 2.3. [16] A nearring (N,+, ·) is said to be non-associative if (N, ·) is nota semigroup.

66 ΘΓ N -group

Definition 2.4. [33] Let N be a nearring and a, b ∈ N . a ≡ b⇔ ∀n ∈ N : na = nb.N is said to be planar nearring if |N/≡| ≥ 3 and if every equation xa = xb+ c (a 6= b)has a unique solution (in N).

Definition 2.5. [39] A double planar nearring (N,+, ∗, ·) is an ordered quadruplewhere each of the ordered triples (N,+, ∗) and (N,+, ·) is a nearring, and where ∗and · are each left distributive over the other. That is, a ∗ (b · c) = (a ∗ b) · (a ∗ c) anda · (b ∗ c) = (a · b) ∗ (a · c), for all a, b, c ∈ N . If each of the nearrings (N,+, ∗) and(N,+, ·) is planar, then (N,+, ∗, ·) is a double planar nearring.

For further concepts in planar nearrings we refer to [2, 40].

3. ΘΓ N-group

Definition 3.1. Let (G,+G) be a group. G is called a ΘΓ N -group if there existsa nearring (N,+, ·) and there exist maps Θ(N × Θ × G → G), Γ(N × Γ ×N → N)containing nearring multiplication ·, ∆Γ(N ×∆Γ × G → G) satisfying the followingconditions.

1. θ is right distributive: (n+m)θg = nθg+Gmθg, for all n,m ∈ N, g ∈ G, θ ∈ Θ;

2. θ is quasi associative: for every n,m ∈ N, γ ∈ Γ, there exists δγ ∈ ∆Γ such that(nγm)θg = nδγ(mθg), for all g ∈ G, θ ∈ Θ.

Example 3.2. Let G = Z6 = {0, 1, 2, 3, 4, 5}. Then (Z6,+6) is a group under additionmodulo 6. Take a nearring N = {0, 2, 4} with + and · defined in Table 1. LetΘ = {θ1, θ2}, Γ = {γ1 = ·, γ2}, ∆Γ = {δγ1 , δγ2} be given by the tables in Figure 1. Itcan be verified that Z6 is a ΘΓ N -group.

+ 0 2 4

0 0 2 4

2 2 4 0

4 4 0 2

· 0 2 40 0 0 0

2 0 2 0

4 0 4 0

Table 1: Binary operations + and · Figure 1: Θ,Γ,∆Γ from Example 3.2


Example 3.3 (Symmetries of a square). It is well known that symmetries of a squareform a group known as the Dihedral groupD4. Consider a square as shown in Figure 2.

Let e denote no change in the square. Let R1 be the rotation of the square by90 degrees, R2 be the rotation by 180 degrees, R3 be the rotation by 270 degrees (allrotations in anti-clockwise direction based on the centroid). Let V be the verticalflip, H be the horizontal flip, D1 and D2 be the diagonal flips.

Take G = {e,R1, R2, R3, V,H,D1, D2}. Then G is a group with the binary oper-ation +G given in Figure 3.

A B

CD

H

V D2 D1

Figure 2: Symmetries of a square Figure 3: Binary operation +G

Let N = (Z8,+8, ·8). Define Θ = {θ1, θ2, θ3}, Γ = {γ1, γ2, γ3} and ∆Γ ={δγ1 , δγ2 , δγ3} as in Figure 5. Take γ1 = ·8.

Using the tables from Figure 5, it can be verified that G is a ΘΓ N -group.

R1: Rotation by 90 degrees

A B

D C

B C

DA

C D

B A

D A

C BR3 = R2 +R1. Rotation by 270 degrees

R2= R1 +R1. Rotation by 180 degrees

R0: Square ABCD with no rotations applied

Figure 4: Some geometrical interpretations of computations

68 ΘΓ N -group

Figure 5: Θ,Γ,∆Γ from Example 3.3

Consider (5γ17)θ1R3 +G (3γ12)θ1R3. This expression is equal to 3θ1R3 +G 6θ1R3.Note that 3θ1R3 is the rotation of the square ABCD by 270 degrees in anti-clockwisedirection repeated three times, which yields R1. Similarly, we obtain 6θ1R3 = R2.Then 3θ1R3 +G 6θ1R3 = R1 +G R2 = R3.

Now let us consider (5γ17 +8 3γ12)θ1R3 = (3 +8 6)θ1R3 = 1θ1R3 = R3. Thus weget, (5γ17 +8 3γ12)θ1R3 = (5γ17)θ1R3 +G (3γ12)θ1R3.

Remark 3.4. Under similar operations, we can show that the group formed by sym-metries of an equilateral triangle is a ΘΓ N -group.


To give the next example of ΘΓ N -group, we require some basic definitions andnotations from lambda calculus. The lambda calculus is a theory of functions asformulas. In this system functions are written as expressions. Lambda calculus wasintroduced by Alonzo Church [17] in 1936 to formalize the concept of effective com-putability. We refer to [20] for the following definitions. The set of λ-terms (notationΛ) is built up from an infinite set of variables V = {v, v′ , v”, . . .} using applicationand (function) abstraction:

x ∈ V → x ∈ Λ, M,N ∈ Λ⇒ (MN) ∈ Λ, M ∈ Λ, x ∈ V ⇒ (λxM) ∈ Λ,where M and N are expressions.

If f and x are lambda terms, and n > 0 a natural number, write fnx for the termf(f(. . . (fx) . . .)), where f occurs n times. For each natural number n, we define alambda term n, called the n-th Church numeral, as n = λfx.fnx. Here are the firstfew Church numerals:

0 = λfx.x, 1 = λfx.fx, 2 = λfx.f(fx), 3 = λfx.f(f(fx)), . . .

The successor function in [34] is defined as + ≡ λwyx.y(wyx) and the product functionis defined as ∗1 ≡ (λxyz.x(yz)).

Example 3.5. Let G = {0, 1, 2, . . .}. We will form a ΘΓ N -group from G. First, weprove that (G, ∗1) is a semigroup.

We show that, (a+ b) ∗1 c = a ∗1 c+ a ∗1 b.Let a, b, c ∈ G. We claim that (a+ b) = λyx.ya+bx. We have

a+ b = λsh.sahSλsh.sbh = (λsh.sah)(λwyx.y(wyx))(λsh.sbh)

= (aλwyx.y(wyx))(λsh.sbh)

= ((a− 1)λwyx.y(wyx))λwyx.y(wyx)(λsh.sbh)= ((a− 1)λwyx.y(wyx))(λyx.y(λsh.sbh)yx)= ((a− 1)λwyx.y(wyx))(λyx.y(yb(x)))= ((a− 1)λwyx.y(wyx))(λyx.yb+1(x)).

Continuing, we get a + b = (λwyx.y(wyx))(λyx.ya+b−1x), and operating once morea+ b = λyx.ya+bx. Clearly, we have a+ b = b+ a.

Now we claim that (a+ b) ∗1 c = λzh.z(a+b)ch. We have(a+ b) ∗1 c = (λsh.sa+bh).(λsh.sch) = (λz.(a+ b)(cz))

= λz.(λsh.sa+bh)((λsh.sch)z = λz.(λsh.sa+bh)(λh.zch)

= λz.(λh.(a+ b)λh.zc(h)h) = λz.(λh.z(a+b)c(h)) = λzh.z(a+b)ch (1)

Now, consider

a ∗1 c+ b ∗1 c = λzh.z(ac)h+ λzh.z(bc)h= (λzh.z(ac)h)(λwyx.y(wyx))(λzh.z(bc)h)

= (ac)(λwyx.y(wyx))(λzh.z(bc)h)

= (ac− 1)(λwyx.y(wyx))(λwyx.y(wyx))(λzh.z(bc)h)

70 ΘΓ N -group

= (ac− 1)(λwyx.y(wyx))(λyx.y(λzh.z(bc)h)yx)= (ac− 1)(λwyx.y(wyx))(λyx.y(y(bc)(x)))= (ac− 1)(λwyx.y(wyx))(λyx.y(bc+1)(x)).

Continuing, we get a ∗1 c+ b ∗1 c = (λwyx.y(wyx))(λyx.y(bc+ac−1)(x)) and operatingonce more

a ∗1 c+ b ∗1 c = (λyx.y(bc+ac)(x)) = (λyx.y(a+b)c(x)). (2)Thus by (1) and (2) we get (a + b) ∗1 c = a ∗1 c + b ∗1 c. Similarly, we geta ∗1 (b+ c) = a ∗1 b+ a ∗1 c.

Now we prove that ∗1 is associative. We have, (a ∗1 b) ∗1 c = (λzh.z(ab)h) ∗1(λzh.z(c)h) = λzh.z(abc)h and a∗1 (b∗1 c) = (λzh.z(a)h)∗1 (λzh.z(bc)h) = λzh.z(abc)h.Hence (a ∗1 b) ∗1 c = a ∗1 (b ∗1 c).

Now, we show that G can be extended to a ΘΓ N -group. To obtain this, Churchpair can be used which is formed by extending Church Numerals to signed numbers.A Church pair contains Church numerals representing a positive and a negative value.Let Ĝ denote the set of signed numbers. On the set Ĝ, addition and subtraction arenaturally defined as follows:

x+ y = [xp, xn] + [yp, yn] = [xp + yp, xn + yn],

x− y = [xp, xn]− [yp, yn] = [xp + yn, xn + yp].Define ~1 as x~1 y = [xp, xn]~1 [yp, yn] = [xp ∗1 yp + xp ∗1 yn, xn ∗1 yp + xn ∗1 yn],for all x, y ∈ Ĝ. Note that (Ĝ, +) is a group. We will show that N=(Ĝ,+, ~1) is anearring. We prove that (x+ y)~1 z = x~1 z + y ~1 z. We have

(x+ y)~1 z = [xp + yp, xn + yn]~1 [zp, zn]

= [(xp + yp) ∗1 zp + (xp + yp) ∗1 zn,(xn + yn) ∗1 zp + (xn + yn) ∗1 zn], (3)

x~1 z + y ~1 z = [xp ∗1 zp + xp ∗1 zn, xn ∗1 zp + xn ∗1 zn]+ [yp ∗1 zp + yp ∗1 zn, yn ∗1 zp + yn ∗1 zn]= [xp ∗1 zp + xp ∗1 zn + yp ∗1 zp + yp ∗1 zn,

xn ∗1 zp + xn ∗1 zn + yn ∗1 zp + yn ∗1 zn]= [(xp + yp) ∗1 zp + (xp + yp) ∗1 zn,

(xn + yn) ∗1 zp + (xn + yn) ∗1 zn]. (4)From (3) and (4) we get (x+ y)~1 z = x~1 z + y ~1 z.

Now, we prove that (x~1 y)~1 z = x~1 (y ~1 z). We have

(x~1 y)~1 z = [xp ∗1 yp + xp ∗1 yn, xn ∗1 yp + xn ∗1 yn] ∗1 [zp, zn]= [(xp ∗1 yp + xp ∗1 yn) ∗1 zp + (xp ∗1 yp + xp ∗1 yn) ∗1 zn,

(xn ∗1 yp + xn ∗1 yn) ∗1 zp + (xn ∗1 yp + xn ∗1 yn) ∗1 zn]= [xp ∗1 yp ∗1 zp + xp ∗1 yn ∗1 zp + xp ∗1 yp ∗1 zn + xp ∗1 yn ∗1 zn,

xn ∗1 yp ∗1 zp + xn ∗1 yn ∗1 zp + xn ∗1 yp ∗1 zn + xn ∗1 yn ∗1 zn]


= [xp ∗1 (yp ∗1 zp + yp ∗1 zn) + xp ∗1 (yn ∗1 zp + yn ∗1 zn),xn ∗1 (yp ∗1 zp + yp ∗1 zn) + xn ∗1 (yn ∗1 zp + yn ∗1 zn)]

= [xp, xn]~1 [yp ∗1 zp + yp ∗1 zn, yn ∗1 zp + yn ∗1 zn]= x~1 (y ~1 z).

Now, define ∗2 ≡ (λxyz.y(xz)) and ~2 asx~2 y = [xp, xn]~2 [yp, yn] = [xp ∗2 yp + xp ∗2 yn, xn ∗2 yp + xn ∗2 yn].

Take Θ = {~2}, Γ = {~1} and ∆Γ = {~1}. We will prove that Ĝ is a ΘΓ N -group.Now, we claim that:

(i) ~2 is right distributive.

(x+ y)~2 z = [xp + yp, xn + yn]~2 [zp, zn]

= [(xp + yp) ∗2 zp + (xp + yp) ∗2 zn,(xn + yn) ∗2 zp + (xn + yn) ∗2 zn], (5)

x~2 z + y ~2 z = [xp ∗2 zp + xp ∗2 zn, xn ∗2 zp + xn ∗2 zn]+ [yp ∗2 zp + yp ∗2 zn, yn ∗2 zp + yn ∗2 zn]= [xp ∗2 zp + xp ∗2 zn + yp ∗2 zp + yp ∗2 zn,

xn ∗2 zp + xn ∗2 zn + yn ∗2 zp + yn ∗2 zn]= [(xp + yp) ∗2 zp + (xp + yp) ∗2 zn,

(xn + yn) ∗2 zp + (xn + yn) ∗2 zn]. (6)From (5) and (6) we have (x+ y)~2 z = x~2 z + y ~2 z.

(ii) ~2 is quasi associative.

(x~1 y)~2 z = [xp ∗1 yp + xp ∗1 yn, xn ∗1 yp + xn ∗1 yn] ∗2 [zp, zn]= [(xp ∗1 yp + xp ∗1 yn) ∗2 zp + (xp ∗1 yp + xp ∗1 yn) ∗2 zn,

(xn ∗1 yp + xn ∗1 yn) ∗2 zp + (xn ∗1 yp + xn ∗1 yn) ∗2 zn]= [xp ∗1 yp ∗2 zp + xp ∗1 yn ∗2 zp + xp ∗1 yp ∗2 zn + xp ∗1 yn ∗2 zn,

xn ∗1 yp ∗2 zp + xn ∗1 yn ∗2 zp + xn ∗1 yp ∗2 zn + xn ∗1 yn ∗2 zn]= [xp ∗1 (yp ∗2 zp + yp ∗2 zn) + xp ∗1 (yn ∗2 zp + yn ∗2 zn),

xn ∗1 (yp ∗2 zp + yp ∗2 zn) + xn ∗1 (yn ∗2 zp + yn ∗2 zn)]= [xp, xn]~1 [yp ∗2 zp + yp ∗2 zn, yn ∗2 zp + yn ∗2 zn]= x~1 (y ~2 z).

Hence (x~1 y)~2 z = x~1 (y ~2 z). Thus Ĝ is a ΘΓ N -group.

Example 3.6. Let (R,+) be the group of real numbers. Take N = (R,+, ·) anda, b, c ∈ R. Define

a div b =

{0 if b = 0,ab if b 6= 0.

Corresponding to the operation div, define divc by a divcb = a div (bc2).

72 ΘΓ N -group

Let Θ = {·, div}, Γ = Θ and ∆Γ = {·, divc}. Clearly, the operations in Θ are rightdistributive. Note that the multiplication operation · in Θ is associative. We have

a divc(b div c) = a div ((b div c)c2) =

abc · c2

= (a div b) div c.

This implies that div is quasi associative. Hence the operations in Θ are quasi asso-ciative. Therefore R is a ΘΓ N -group.

Proposition 3.7. 1. A non associative nearring induced by a nearring forms aΘΓ N -group.

2. A double planar nearring induced by a nearring forms a ΘΓ N -group.

Proof. 1. Let (N,+, .) be a nearring and k : N → End(N,+) be a mapping.Define ∗ : N × N → N by a ∗ b = k(b)(a).b = f(a).b, where f = k(b) is anendomorphism for each b. Then (N,+, ∗) is a non associative nearring. For,

(a+ b) ∗ c = k(c)(a+ b).c = f(a+ b).c = (f(a).c+ f(b).c) = (a ∗ c) + (b ∗ c).

Let f(k) ∈ End(N,+, .) be such that for a, b, c ∈ N a ∗f(k) b = f(k)(a).b. Wewill prove: (a ∗ b) ∗ c = a ∗k(c)◦k(b) (b ∗ c). We have(a ∗ b) ∗ c = (k(b)(a).b) ∗ c = k(c)(k(b)(a).b).c = [k(c)k(b)(a).k(c)(b)].c

= [k(c)k(b)(a)].[k(c)(b).c] = [k(c) ◦ k(b)](a).(b ∗ c) = a ∗k(c)◦k(b) (b ∗ c).Hence N forms a ΘΓ N -group with Θ = {∗}, Γ = {∗}, and ∆Γ = {∗k(c)◦k(b)}.

2. Let N be a nearring. Define a ∗ b = a |b| and

a ◦ b ={

0 if b = 0,a b|b| if b 6= 0.

Then (N,+, ∗), (N,+, ◦) are planar nearrings. We have (a∗b)◦c = (a◦c)∗(b◦c),(a ◦ b) ∗ c = (a ∗ c) ◦ (b ∗ c). Now, (N,+, ∗, ◦) is a double planar nearring. Defineaδc◦b = (a∗c)◦(b) and aδc∗b = (a◦c)∗b. Now, (a◦b)∗c = (a∗c)◦(b∗c) = aδc◦(b∗c),and (a ∗ b) ◦ c = (a ◦ c) ∗ (b ◦ c) = aδc∗(b ◦ c). Hence N forms a ΘΓ N -group withΘ = {∗, ◦}, Γ = {∗, ◦}, and ∆Γ = {δc∗, δc◦}.

Proposition 3.8. 1. Every N -group is a ΘΓ N -group.

2. Every gamma nearring is a ΘΓ N -group.

Proof. 1. Take Θ = {θ}, Γ = {·} and ∆Γ = {θ}.

2. Take N = G, Θ = Γ and ∆Γ = Γ.

Proposition 3.9. Let G be a group and N be nearring. Then for all g ∈ G, n ∈ N :

1. 0Nθg = 0G, for all θ ∈ Θ.

2. (−n)θg = −nθg, for all θ ∈ Θ.

3. For γ ∈ Γ, nγ0N = 0N ⇒ nδγ0G = 0G.


4. Let N = Nc. Then for γ ∈ Γ, θ ∈ Θ, (nγm)θg = nδγ0G.

Proof. 1. (0N + 0N )θg = 0Nθg + 0Nθg. Then 0Nθg = 0Nθg + 0Nθg. Hence0Nθg = 0G.

2. 0G = 0Gθg = (−n+ n)θg = (−n)θg + nθg. This gives (−n)θg = −nθg.

3. (nγ0N )θg = nδγ(0Nθg) = nδγ0G =⇒ nδγ0G = (nγ0N )θg = 0Nθg = 0G.

4. (nγm)θg = (nγ0Nγm)θg = (nγ0N )θg = nδγ(0Nθg) = nδγ0G.

Definition 3.10. Let G be a ΘΓ N -group. A subgroup (H,+) of (G,+) is said tobe a ΘΓ N -subgroup of G if NΘH ⊆ H.

Definition 3.11. Let N be a nearring and G, G′

be ΘΓ N -groups. Then h : G→ G′

is called a ΘN -homomorphism if it satisfies

1. h(x+ y) = h(x) + h(y) and

2. h(nθx) = nθh(x) for all n ∈ N , x, y ∈ G and θ ∈ Θ.

The set of all ΘN -homomorphisms is denoted by HomΘ(G,G′).

Definition 3.12. Ker h = {x ∈ G|h(x) = 0′}.

Definition 3.13. A normal subgroup H of a ΘΓ N -group (G,+) is called a ΘN -idealof G if nθ(x+ a)− nθx ∈ H for all n ∈ N , x ∈ G, a ∈ H and θ ∈ Θ.

Remark 3.14. Let H be a ΘΓ N -subgroup of (G,+). Then the following two condi-tions are equivalent:

1. H is a Θ-N ideal of the ΘΓ-N group G; and

2. x ≡ y(mod H), a ≡ b(mod H)⇒ x+a ≡ y+b(mod H), and nθx ≡ nθy(mod H).

Verification:1 ⇒ 2: Suppose that x1 ≡ x

′

1(mod H) and x2 ≡ x′

2(mod H). This impliesthat x1 − x

′

1 ∈ H,x2 − x′

2 ∈ H. Now we show that x1 + x2 ≡ x′

1 + x′

2(mod H)and nθx1 ≡ nθx

′

1(mod H). Now (x1 + x2) − (x′

1 + x′

2) = x1 + (x2 − x′

2) − x′

1 =x1 + (x2 − x

′

2) + x1 − x1 − x′

1 = (x1 + (x2 − x′

2) − x1) − (x′

1 − x1) ∈ H (since His normal, and x2 − x

′

2 ∈ H). This implies (x1 + x2) ≡ (x′

1 + x′

2)(mod H). Nownθx1 − nθx

′

1 = nθ(x1 − x′

1 + x′

1) − nθx′

1 ∈ H (since x1 − x′

1 ∈ H and H is an idealof G). This means that nθx1 ≡ nθx

′

1(mod H).2 ⇒ 1: First we show that H is a normal subgroup of G. Let x ∈ G and h ∈ H.

We know that x ≡ x(mod H) and h ≡ 0(mod H). By the assumed condition,x + h ≡ x + 0(mod H). This implies x + h ≡ x(mod H). Thus x + h − x ∈ H. Letn ∈ N . We know that n ≡ n(mod H) and x + h ≡ x(mod H). By the assumedcondition, nθ(x+ h) ≡ nθx(mod H). This implies that nθ(x+ h)− nθx ∈ H. HenceH is an ideal of G.

74 ΘΓ N -group

Remark 3.15. Let G be a ΘΓ N -group and H a normal subgroup of (G,+).Then the following two conditions are equivalent:

1. nθ(x+ a)− nθx ∈ H, for all n ∈ N, x ∈ G, a ∈ H and θ ∈ Θ, and

2. nθ(b+ x)− nθx ∈ H, for all n ∈ N, x ∈ G, b ∈ H and θ ∈ Θ.

Verification:1⇒ 2 : nθ(b+ x)− nθx = nθ(x− x+ b+ x)− nθx = nθ(x+ a)− nθx ∈ H (by 1).The proof of 2⇒ 1 is similar.

Proposition 3.16. If I EΘN G then G/I = {g + I | g ∈ G} is a ΘΓ N -group.

Proof. First, we define operations +, θ on G/I as follows:

(g1 + I) + (g2 + I) = (g1 + g2) + I, nθ(g1 + I) = nθg1 + I.

It is easy to show that + is well-defined.We will prove that θ is well-defined. Let nθ(g1 +I) = nθ(g

′1 +I) and x ∈ nθg1 +I.

Then x = nθg1+i = nθ(g1+0G)+i = nθ(g′1+i

′1)+i = i

′+nθg′1+i = 0G+i′+nθg′1+i =

(nθg′1 − nθg′1) + i′ + nθg′1 + i = nθg′1 + (−nθg′1 + i′ + nθg′1) + i ∈ nθg′1 + I. Hencenθg1 + I ⊆ nθg′1 + I. Similarly, nθg′1 + I ⊆ nθg1 + I. Hence nθg1 + I = nθg′1 + I.

To show that G/I is a ΘΓ N -group, we will prove for n,m ∈ N, γ ∈ Γ, thereexists δγ ∈ ∆Γ such that (nγm)θ(g+I) = nδγ(mθ(g+I)). Consider (nγm)θ(g+I) =((nγm)θg) + I. Then there exists δγ ∈ ∆Γ such that (nγm)θg = nδγ(mθg). Then(nδγ(mθg)) + I = nδγ(mθg + I) = nδγ(mθ(g + I)). Therefore (nγm)θ(g + I) =nδγ(mθ(g + I)). Hence θ is quasi associative.

Clearly θ is right distributive. Hence G/I is a ΘΓ N -group. �

Definition 3.17. Let I EΘN G. Then G/I = {g + I | g ∈ G} is called a factor ΘΓN -group.

Proposition 3.18. Let f : G → G′ be a ΘN -homomorphism. Then Ker f is aΘN -ideal of G. Conversely, every ΘN -ideal is the kernel of a ΘN -homomorphism.

Proof. We have f(0) = 0′. Hence 0 ∈ Ker f . Let g ∈ G,n ∈ N, a ∈ Ker f . Thenf(a) = 0′. Now,

f(g + a− g) = f(g) + f(a)− f(g) = 0′ ⇒ g + a− g ∈ Ker f,f(nθ(x+ a)− nθx) = f(nθ(x+ a))− f(nθx) = nθf(x+ a)− nθf(x)

= nθ(f(x) + f(a))− nθf(x) = 0′ ⇒ nθ(x+ a)− nθx ∈ Ker f.Hence Ker f is a ΘN -ideal of G. To prove the converse, define φ : G → G/I byφ(g) = g + I. We prove that φ is well defined and one-one. We have, g1 = g2⇔ g1 + I = g2 + I ⇔ φ(g1) = φ(g2). Let g+ I ∈ G/I. Then φ(g) = g+ I. Hence φ isonto. φ is a homomorphism because

φ(g1 + g2) = (g1 + g2) + I = g1 + I + g2 + I = φ(g1) + φ(g2),

φ(nθg) = nθg + I = nθ(g + I) = nθφ(g).

Now, Ker φ = {x ∈ G | φ(x) = 0+I} = {x ∈ G | g+I = 0+I} = I. Hence ΘN -Idealis the kernel of a ΘN -homomorphism. �


Theorem 3.19. Let f : G → G′ be an onto Θ N -homomorphism and K = kerf .Then K is an ideal of G and G/K ∼= G′.

Proof. Define φ : G/K → G′ by φ(a+K) = f(a). We will show that φ is well definedand one-one. Let a, b ∈ G. We have a+K = b+K ⇔ a− b ∈ K ⇔ f(a− b) = 0⇔f(a) − f(b) = 0 ⇔ f(a) = f(b) ⇔ φ(a + K) = φ(b + K). Now, we prove that φ isonto. Let y ∈ G′. As f is onto, y = f(a) for some a ∈ G. Now a + K ∈ G/K andφ(a+K) = f(a) = y. Now we prove that φ is homomorphism. We have

φ((a+K) + (b+K)) = φ((a+ b) +K) = f(a+ b) = f(a) + f(b)

= φ(a+K) + φ(b+K),

φ(nθ(a+K)) = φ(nθa+K) = f(nθa) = nθf(a) = nθφ(a+K).

Thus G/K ∼= G′. �

Theorem 3.20. 1. Let f : G → G′ be an onto Θ N -homomorphism and H =Ker f . If K ′ is ΘΓ N -subgroup (resp. ΘN -ideal) of G and K = {x ∈ G :f(x) ∈ K ′} = f−1(K ′), then K is a ΘΓ N -subgroup(resp. ΘN -ideal) of G byH ⊆ K and G/K ∼= G′/K ′.

2. Let H and K be ΘN -ideals of ΘΓ N -group G by H ⊆ K. Then G/K ∼=(G/H)/(K/H).

Proof. 1. Define φ : G → G′/K ′ by φ(x) = f(x) + K ′. First, we show that φis a homomorphism. Let a, b ∈ G. We have φ(a + b) = f(a + b) + K ′ =(f(a) + f(b)) +K ′. We claim (f(a) + f(b)) +K ′ = (f(a) +K ′) + (f(b) +K ′).Note that x ∈ (f(a)+f(b))+K ′ ⇒ x = f(a)+f(b)+k′1 = f(a)+0+f(b)+k′1 ⇒x ∈ f(a) +K ′ + f(b) +K ′ ⇒ (f(a) + f(b)) +K ′ ⊆ (f(a) +K ′) + (f(b) +K ′).Let y ∈ (f(a) + K ′) + (f(b) + K ′). Hence there exist k′1, k′2 ∈ K ′ such thaty = f(a) + k′1 + f(b) + k

′2. Then y = f(a) + f(b) − f(b) + k′1 + f(b) + k′2

∈ f(a) + f(b) + K ′ ⇒ (f(a) + K ′) + (f(b) + K ′) ⊆ (f(a) + f(b)) + K ′. Henceφ(a+ b) = (f(a) + f(b)) +K ′ = (f(a) +K ′) + (f(b) +K ′) = φ(a) + φ(b). Now,φ(nθa) = nθf(a) + K ′ = nθ(f(a) + K ′) = nθφ(a). Now we prove that φ isonto. Let a′ + K ′ ∈ G′/K ′, a′ ∈ G′. Since f is onto, there exists a ∈ G suchthat f(a) = a′. Hence φ(a) = f(a) + K ′ = a′ + K ′. By Theorem 3.18, we getG/Ker φ ∼= G′/K ′. Now, we haveKer φ = {x | φ(x) = e+K ′} = {x | f(x)+K ′ = e+K ′} = {x | f(x) ∈ K ′} = K.Let x ∈ G, k ∈ K,n ∈ N . As φ(k) = 0, we get

φ(x+ k − x) = φ(x) + φ(k)− φ(x) = 0⇒ x+ k − x ∈ K;φ(nθk) = nθφ(k) = nθ0 = 0⇒ nθk ∈ K;

φ(nθ(x+ k)− nθx) = φ(nθ(x+ k))− φ(nθx)= nθ(φ(x) + φ(k))− nθφ(x) = 0⇒ nθ(x+ k)− nθx ∈ K.

Hence K is ΘΓ N -subgroup (resp. ΘN -ideal) of G and G/K ∼= G′/K ′. Letx ∈ H. Then we have, f(x) = e′ ∈ K ′. This implies φ(x) = f(x) + K ′ =e′ +K ′ = K ′. We get x ∈ Ker φ = K. Hence H ⊆ K.

76 ΘΓ N -group

2. H is a normal subgroup of G and K is a normal subgroup of G containing H.Hence K/H and G/H are factor ΘΓ N -groups.

First, we prove that K/H is a normal subgroup of G/H. Define f : G/H →G/K by f(x+H) = x+K. We prove that f is well-defined. We have x+H =y +H ⇒ x− y ∈ H ⇒ x− y ∈ K ⇒ x+K = y +K ⇒ f(x+H) = f(y +H).We now show that f is an onto homomorphism with Ker f = K/H. We have

f [(x+H) + (y +H)] = f [(x+ y) +H] = (x+ y) +K

= (x+K) + (y +K) = f(x+H) + f(y +H), and

f(nθ(x+H)) = f(nθx+H) = nθx+K = nθ(x+K) = nθf(x+H).

Let x+K ∈ G/K. Then there exists x+H ∈ G/H such that f(x+H) = x+K.We have, Ker f = {g+H|f(g+H) = e+K} = {g+H|g ∈ K} = K/H. HenceK/H is a normal subgroup of G/H. By Theorem 3.19, (G/H)/Ker f ∼= G/K.Thus, (G/H)/(K/H) ∼= G/K.

�

4. Conclusion

We have introduced the algebraic structure of ΘΓ N -group as a natural extensionof N -group and gamma nearring. We have shown that the lambda calculus sys-tem induces a ΘΓ N -group. Other examples of ΘΓ N -group include non associativenearrings and double planar nearrings. We have defined the concept of ideal of ΘΓN -group and proved isomorphism theorems. Different prime ideal notions and corre-sponding radicals of ΘΓ N -group can be studied as future work.

Acknowledgement. The authors thank the anonymous referee for his/her com-ments and suggestions. All authors acknowledge Manipal Institute of Technology,Manipal University for their encouragement. The first author acknowledges ManipalUniversity for the financial support Dr. T. M. A. Pai MU-PhD scholarship.

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(received 07.04.2017; in revised form 25.07.2017; available online 06.10.2017)

Department of Mathematics, Manipal Institute of Technology, Manipal University, Manipal,Karnataka 576104, India.

E-mail: [email protected]





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1. Introduction · to a Kurosh-Amitsur prime radical for nearrings (see [13]). Veljko [37,38] gave...

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